We demonstrate the remarkable effectiveness of boundary value formulationscoupled to numerical continuation for the computation of stable and unstablemanifolds in systems of ordinary differential equations. Specifically, weconsider the Circular Restricted Three-Body Problem (CR3BP), which models themotion of a satellite in an Earth- Moon-like system. The CR3BP has manywell-known families of periodic orbits, such as the planar Lyapunov orbits andthe non-planar Vertical and Halo orbits. We compute the unstable manifolds ofselected Vertical and Halo orbits, which in several cases leads to thedetection of heteroclinic connections from such a periodic orbit to invarianttori. Subsequent continuation of these connecting orbits with a suitable endpoint condition and allowing the energy level to vary, leads to the furtherdetection of apparent homoclinic connections from the base periodic orbit toitself, or the detection of heteroclinic connections from the base periodicorbit to other periodic orbits. Some of these connecting orbits could be ofpotential interest in space-mission design.
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